DECEMBER 2010
LVOV ET AL.
2605
Oceanic Internal-Wave Field: Theory of Scale-Invariant Spectra
YURI V. LVOV
Rensselaer Polytechnic Institute, Troy, New York
KURT L. POLZIN
Woods Hole Oceanographic Institution, Woods Hole, Massachusetts
ESTEBAN G. TABAK
New York University, New York, New York
NAOTO YOKOYAMA
Doshisha University, Kyotanabe, Japan
(Manuscript received 28 August 2008, in final form 28 July 2010)
ABSTRACT
Steady scale-invariant solutions of a kinetic equation describing the statistics of oceanic internal gravity
waves based on wave turbulence theory are investigated. It is shown in the nonrotating scale-invariant limit
that the collision integral in the kinetic equation diverges for almost all spectral power-law exponents. These
divergences come from resonant interactions with the smallest horizontal wavenumbers and/or the largest
horizontal wavenumbers with extreme scale separations.
A small domain is identified in which the scale-invariant collision integral converges and numerically find
a convergent power-law solution. This numerical solution is close to the Garrett–Munk spectrum. Power-law
exponents that potentially permit a balance between the infrared and ultraviolet divergences are investigated.
The balanced exponents are generalizations of an exact solution of the scale-invariant kinetic equation, the
Pelinovsky–Raevsky spectrum. A small but finite Coriolis parameter representing the effects of rotation is
introduced into the kinetic equation to determine solutions over the divergent part of the domain using
rigorous asymptotic arguments. This gives rise to the induced diffusion regime.
The derivation of the kinetic equation is based on an assumption of weak nonlinearity. Dominance of the
nonlocal interactions puts the self-consistency of the kinetic equation at risk. However, these weakly nonlinear stationary states are consistent with much of the observational evidence.
1. Introduction
Wave–wave interactions in continuously stratified fluids
have been a subject of intensive research in the last few
decades. Of particular importance is the observation of
a nearly universal internal-wave energy spectrum in the
ocean, first described by Garrett and Munk (Garrett and
Munk 1972, 1975; Cairns and Williams 1976; Garrett and
Munk 1979). However, it appears that ocean is too complex to be described by one universal model. Accumulating
Corresponding author address: Naoto Yokoyama, Department
of Aeronautics and Astronautics, Graduate School of Engineering,
Kyoto University, Kyoto 606-8501, Japan.
E-mail: yokoyama@kuaero.kyoto-u.ac.jp
DOI: 10.1175/2010JPO4132.1
! 2010 American Meteorological Society
evidence suggests that there is measurable variability
of observed experimental spectra (Polzin and Lvov 2010,
manuscript submitted to Rev. Geophys., hereafter PL).
In particular, we have analyzed the decades of observational programs, and we have come to the conclusion
that the high-frequency–high-wavenumber part of the
spectrum can be characterized through simple powerlaw fits with variable exponents.
It is generally thought (Müller et al. 1986; Olbers 1974,
1976; PL) that nonlinear interactions significantly contribute to determining the background oceanic spectrum,
and this belief motivates the investigation of spectral
evolution equations for steady balances. A particularly
important study in this regard is the demonstration that
the Garrett–Munk (GM) vertical wavenumber spectrum
2606
JOURNAL OF PHYSICAL OCEANOGRAPHY
is stationary and supports a constant downscale energy
flux (McComas and Müller 1981) associated with resonant interactions. The accumulating evidence alluded to
above suggests there is more to the story.
The purpose of the present study is to lay down a firm
theoretical framework that allows a detailed analysis of
power-law spectra of internal waves in the ocean. We
investigate the parameter space of the possible power
laws with a specific focus on extreme scale-separated interactions and their role in dominating spectral transfers.
We then use this theoretical framework to interpret the
observed oceanic variability.
›np
›t
5 4p
with
"
dp12 jV pp
ð
1 ,p2
f pp
1 p2
j2 f pp
1 p2
dp!p
1 !p2
dv
p !vp1 !vp2
p
Because of the quadratic nonlinearity of the underlying fluid equations and dispersion relation allowing
three-wave resonances, internal waves interact through
triads. In the weakly nonlinear regime, the nonlinear
interactions among internal waves concentrate on their
resonant set and can be described by a kinetic equation,
which assumes the familiar form (Caillol and Zeitlin 2000;
Hasselmann 1966; Kenyon 1966, 1968; Lvov and Tabak
2001, 2004; McComas and Bretherton 1977; Milder 1990;
Müller and Olbers 1975; Olbers 1974, 1976; Pelinovsky
and Raevsky 1977; Pomphrey et al. 1980; Voronovich
1979; Zakharov et al. 1992)
p
! jV p12 ,p j2 f p12 p dp !p !p dv
1
5 np np ! np (np 1 np ).
1
2
1
2
p
p1!vp2!vp
p
! jV p2,p1j2 f pp2 1 dp !p!p dv
2
1
p2!vp!vp1
#
,
(1)
2
Here np 5 n(p) is a three-dimensional (3D) action
spectrum [see Eq. (16)] with wavenumber p 5 (k, m):
that is, k and m are the horizontal and vertical components of p. Action or wave action can be viewed as
‘‘number’’ of waves with a given wavenumber. The frequency vp is given by a linear dispersion relation (12)
below. Consequently, wave action multiplied by frequency vpnp can be seen as quadratic spectral energy
density of internal waves. Note that the wavenumbers
are oppositely signed variables, whereas the wave frequencies are always positive. The factor V pp p is the in1 2
teraction matrix element describing the transfer of wave
action among the members of a triad composed of three
wave vectors p 5 p1 1 p2.
Following Kolmogorov’s viewpoint of energy cascades
in isotropic Navier–Stokes turbulence, one may look for
statistically stationary states using scale-invariant solutions
to the kinetic Eq. (1). The solution may occur in an inertial subrange of wavenumbers and frequencies that
are far from those where forcing and dissipation act
and also are far from characteristic scales of the system,
including the Coriolis frequency resulting from the rotation of the earth, the buoyancy frequency due to stratification and the ocean depth. Under these assumptions, the
dispersion relation and the interaction matrix elements
are locally scale invariant. It is natural, therefore, in this
restricted domain, to look for self-similar solutions of
Eq. (1), which take the form
n(k, m) 5 jkj!a jmj!b .
VOLUME 40
(2)
Values of a and b for which the right-hand side of Eq. (1)
vanishes identically correspond to steady solutions of
the kinetic equation and we hope also to statistically
steady states of the ocean’s wave field. Unlike Kolmogorov turbulence, the exponents that give steady solutions cannot be determined by the dimensional analysis
alone (see, e.g., Polzin 2004). This is the case because
of multiple characteristic length scales in anisotropic
systems.
Before seeking steady solutions, however, one should
find out whether the improper integrals1 in the kinetic
Eq. (1) converge. This is related to the question of
locality of the interactions: a convergent integral characterizes the physical scenario where interactions of
neighboring wavenumbers dominate the evolution of
the wave spectrum, whereas a divergent one implies
that distant, nonlocal interactions in the wavenumber
space dominate.
In the present paper, we demonstrate analytically that
the internal-wave collision integral diverges for almost
all values of a and b. In particular, the collision integral
has an infrared (IR) divergence at 0 (i.e., jk1j or jk2j / 0)
and an ultraviolet (UV) divergence at infinity (i.e., jk1j
and jk2j / ‘). Thus, IR divergence comes from interactions with the smallest wavenumber, and UV divergence comes from interactions with the largest horizontal
wavenumber. There is only one exception where the integral converges: the segment with b 5 0 and 7/ 2 , a , 4.
The b 5 0 line corresponds to wave action independent of
Ðb
Ðb
Improper
integralsÐhave the form lim a f (x) dx, lim a f (x) dx,
Ðc
a!!‘
b!‘
b
lim
f (x) dx, or lim1 c f (x) dx. When the limit exists (and is
c!b! a
c!a
a number), the improper integral is called convergent; when the
limit does not exist or is infinite, the improper integral is called
divergent.
1
DECEMBER 2010
LVOV ET AL.
vertical wavenumbers, ›n/›m 5 0. Within this segment,
we numerically determine a new steady convergent solution to Eq. (1), with
n(k, m) } jkj!3.7 .
(3)
This solution is not far from the large-wavenumber form
of the GM spectrum (Garrett and Munk 1972, 1975;
Cairns and Williams 1976; Garrett and Munk 1979),
n(k, m) } jkj!4 .
(4)
Alternatively, one can explore the physical interpretation of divergent solutions. We find a region in (a, b)
space where there are both IR and UV divergences
having opposite signs. This suggests a possible scenario
where the two divergent contributions may cancel each
other, yielding a steady state. An example of such a case
is provided by the Pelinovsky–Raevsky (PR) spectrum
(Pelinovsky and Raevsky 1977),
nk,m } jkj!7/2 jmj!1/2 .
(5)
This solution, however, is only one among infinitely
many. The problem at hand is a generalization of the
concept of principal value integrals: for a and b, which
give opposite signs of the divergences at zero and infinity, one can regularize the integral by cutting out
small neighborhoods of the two singularities in such a
way that the divergences cancel each other and the remaining contributions are small. Hence, all the exponents that yield opposite-signed divergence at both ends
can be steady solutions of Eq. (1). As we will see below
this general statement helps to describe the experimental
oceanographic data that are available to us. The nature of
such steady solutions depends on the particular truncation
of the divergent integrals.
So far, we have kept the formalism at the level of the
self-similar limit of the kinetic Eq. (1). However, once
one considers energy transfer mechanisms dominated
by interactions with extreme modes of the system, one
can no longer neglect the deviations from self-similarity
near the spectral boundaries: the inertial frequency
due to the rotation of the earth at the IR end and the
buoyancy frequency and/or dissipative cutoffs at the
UV end.
For example, we may consider a scenario in which
interactions with the smallest horizontal wavenumbers
dominate the energy transfer within the inertial subrange, either because the collision integral at infinity
converges or because the system is more heavily truncated
2607
at the large wavenumbers by wave breaking or dissipation. We will demonstrate that the IR divergence
of the collision integral has a simple physical interpretation: the evolution of each wave is dominated
by the interaction with its nearest neighboring vertical wavenumbers, mediated by the smallest horizontal wavenumbers of the system. Such a mechanism
is called induced diffusion (ID) in the oceanographic
literature.
To bring back the effects of the rotation of the earth
in Eq. (1), one introduces the Coriolis parameter f there
and in the linear dispersion relation. Because we are
considering the evolution of waves with frequency v
much larger than f, f can be considered to be small.
However, because the interaction with waves near f
dominates the energy transfer, one needs to invert the
order in which the limits are taken, postponing making
f 5 0 to the end. This procedure gives rise to an integral
that diverges like f raised to a negative power smaller
than 21 but multiplied by a prefactor that vanishes
if either 9 2 2a 2 3b 5 0 or b 5 0. These are the
induced diffusion lines of steady-state solutions, found
originally in McComas and Müller (1981) as a diffusive
approximation to the kinetic equation. This family of
stationary states does a reasonable job of explaining the
gamut of observed variability. The rigorous asymptotic
analysis presented here clearly implies the induced
diffusion family of stationary states makes sense only in
the IR divergent subdomain of (a, b) space, and we find
that the data are located in this subdomain.
The present paper investigates in detail the parameter space (a, b) of a general power-law spectrum (2),
compares this parameter space to the ocean observations, and gives possible interpretation. Furthermore,
the present study places the previously obtained ID
curves (McComas and Müller 1981) and Pelinovsky–
Raevsky spectrum (Pelinovsky and Raevsky 1977) into
a much wider context. Finally, we present a general
theoretical background that we are going to exploit for
future studies.
The paper is organized as follows: Wave turbulence
theory for the internal wave field and the corresponding kinetic equation are briefly summarized in section 2
along with the motivating observations. We analyze the
divergence of the kinetic equation in section 3. Section 4
includes a special convergent power-law solution that
may account for the GM spectrum. In section 5, we introduce possible quasi-steady solutions of the kinetic
equation that are based on cancellations of two singularities. Section 6 shows that the IR divergence is dominated by induced diffusion, and we compute the family
of power-law solutions that arises from taking it into
account. We conclude in section 7.
2608
JOURNAL OF PHYSICAL OCEANOGRAPHY
2. Wave turbulence theory for internal waves
a. Background and history
The idea of using wave turbulence formalism to describe internal waves is certainly not new; it dates back
to Kenyon (1966, 1968), with calculations of the kinetic
equations for oceanic spectra presented in Olbers (1976),
McComas and Bretherton (1977), and Pomphrey et al.
(1980). Various formulations have been developed for
characterizing wave–wave interactions in stratified wave
turbulence in the last four decades (Y.V. Lvov, K. L.
Polzin, and N. Yokoyama 2007, unpublished manuscript, hereafter LPY; for details, see Caillol and Zeitlin
2000; Hasselmann 1966; Kenyon 1966, 1968; Lvov and
Tabak 2001, 2004; McComas and Bretherton 1977;
Milder 1990; Müller and Olbers 1975; Olbers 1974, 1976;
Pelinovsky and Raevsky 1977; Pomphrey et al. 1980;
Voronovich 1979). We briefly discuss the derivation of
the kinetic equation and wave–wave interaction matrix
elements below in Eq. (15).
The starting point for the most extensive investigations
has been a noncanonical Hamiltonian formulation in
Lagrangian coordinates (McComas and Müller 1981) that
requires an unconstrained approximation in smallness
of wave amplitude in addition to the assumption that
nonlinear transfers take place on much longer time
scales than the underlying linear dynamics. Other work
has as its basis a formulation in Clebsch-like variables
(Pelinovsky and Raevsky 1977) and a non-Hamiltonian
formulation in Eulerian coordinates (Caillol and Zeitlin
2000). Here, we employ a canonical Hamiltonian representation in isopycnal coordinates (Lvov and Tabak
2001, 2004), which, as a canonical representation, preserves the original symmetries and hence conservation
properties of the original equations of motion.
Energy transfers in the kinetic equation are characterized by three simple mechanisms identified by McComas
and Bretherton (1977) and reviewed by Müller et al.
(1986). These mechanisms represent extreme scaleseparated limits. One of these mechanisms represents
the interaction of two small-vertical-scale, high-frequency
waves with a large-vertical-scale, near-inertial (frequency
near f ) wave and is called induced diffusion (ID). The
ID mechanism exhibits a family of stationary states: that
is, a family of solutions to Eq. (2). A comprehensive inertial-range theory with constant downscale transfer of
energy can be obtained by patching these mechanisms
together in a solution that closely mimics the empirical
universal spectrum (GM; McComas and Müller 1981).
A fundamental caveat from this work is that the
interaction time scales of high-frequency waves are
sufficiently small at small spatial scales as to violate the
assumption of weak nonlinearity.
VOLUME 40
In parallel work, Pelinovsky and Raevsky (1977) derived a kinetic equation for oceanic internal waves. They
also have found the statistically steady-state spectrum
of internal waves, Eq. (5), which we propose to call the
Pelinovsky–Raevsky spectrum. This spectrum was later
found in Caillol and Zeitlin (2000) and Lvov and Tabak
(2001, 2004). It follows from applying the Zakharov–
Kuznetsov conformal transformation (Zakharov et al.
1992), which effectively establishes a map between
the very large and very small wavenumbers. Making
these two contributions cancel pointwise yields the
solution (5).
Both Pelinovsky and Raevsky (1977) and Caillol and
Zeitlin (2000) noted that the solution (5) comes through
a cancellation between oppositely signed divergent contributions in their respective collision integrals. A fundamental caveat is that one cannot use conformal mapping
for divergent integrals. Therefore, the existence of such
a solution is fortuitous.
Here, we demonstrate that our canonical Hamiltonian
structure admits a similar characterization: power-law
solutions of the form (2) return collision integrals that
are, in general, divergent. Regularization of the integral
allows us to examine the conditions under which it is
possible to rigorously determine the power-law exponents (a, b) in Eq. (2) that lead to stationary states. In
doing so, we obtain the ID family.
The situation is somewhat peculiar: We have assumed
weak nonlinearity to derive the kinetic equation. The
kinetic equation then predicts that nonlocal, strongly
scale-separated interactions dominate the dynamics.
These interactions have less chance to be weakly nonlinear than regular, ‘‘local’’ interactions. Thus, the derivation of the kinetic equation and its self-consistency
are at risk. In our subsequent work (LPY), we provide
a possible resolution of this puzzle. However, as we will
see below, despite this caveat, the weakly nonlinear
theory is consistent with much of the observational
evidence.
b. Experimental motivation
Power laws provide a simple and intuitive physical
description of complicated wave fields. Therefore, we
assumed that the spectral energy density can be represented as Eq. (2) and undertook a systematic study of
published observational programs. In doing so, we were
fitting the experimental data available to us by powerlaw spectra. We assume that the power laws offer a good
fit of the data in the high-frequency, high-wavenumber
parts of the spectrum. We do not assume that spectra are
given by Garrett–Munk spectrum.
DECEMBER 2010
In most instances, vertical wavenumber and frequency
power laws were estimated by superimposing best-fit
lines on top of one-dimensional spectra. The quoted
power laws are the asymptotic relations of these best-fit
lines. Fits in the frequency domain included only periods smaller than 10 h, thereby eliminating the inertial
peak and semidiurnal tides from consideration. For onedimensional spectra, there is an implicit assumption
that the high-frequency–high-wavenumber spectra are separable. Because both 1D spectra are red, frequency spectra
are typically dominated by low-vertical-wavenumber motions, and vertical wavenumber spectra are dominated by
low frequencies. Care should be taken to distinguish these
results from the high-wavenumber asymptotics of truly
two-dimensional spectra. We have included two realizations of two-dimensional displacement spectra in isopycnal
coordinates [the Patches Experiment (PATCHEX2) and
the Surface Wave Process Program (SWAPP)]. The quoted
power laws in these instances were estimated using a
straight edge and x-by-eye procedure.
Below we list extant datasets with concurrent vertical
profile and current meter observations and some major experiments utilizing moored arrays, along with our
best estimate of their high-wavenumber, high-frequency
asymptotics:
d
d
d
d
d
d
d
d
d
d
2609
LVOV ET AL.
Site-D (Foffonoff 1969; Silverthorne and Toole 2009):
energy spectra are m22.0 and v22.0;
the Frontal Air-Sea Interaction Experiment (FASINEX;
Weller et al. 1991; Eriksen et al. 1991): energy spectra
are m22.3 and v21.85;
the Internal Wave Experiment (IWEX; Müller et al.
1978): energy spectrum is 2k22.460.4v21.75;
the Salt Finger Tracer Release Experiment (SFTRE;
Schmitt et al. 2005)/Polymode IIIc (PMIII; Keffer
1983): energy spectra are m22.4 and v21.9;
the North Atlantic Tracer Release Experiment
(NATRE; Polzin et al. 2003)–subduction (Weller et al.
2004): energy spectra are m22.55 (observed: NATRE1)
or m22.75 (minus vortical contribution: NATRE2) and
v21.35;
PATCHEX1 (Gregg et al. 1993; Chereskin et al. 2000):
energy spectra are m21.75 and v21.65 2 v22.0;
PATCHEX2 (Sherman and Pinkel 1991): energy spectrum is m21.75v21.65 2 m21.75v22.0;
the SWAPP experiment (Anderson 1992): energy spectrum is m21.9v22.0.
the Storm Transfer and Response Experiment (STREX;
D’Asaro 1984)/Ocean Storms Experiment (OS;
D’Asaro 1995): energy spectra are m22.3 and v22.2;
the Midocean Acoustic Transmission Experiment
(MATE; Levine et al. 1986): energy spectra are 2m22.1
and v21.7; and
d
the Arctic Internal Wave Experiment (AIWEX; Levine
et al. 1987; D’Asaro and Morehead 1991): energy spectra are m22.25 and v21.2.
Two estimates of the NATRE spectrum are provided:
NATRE1 represents the observed spectrum, and NATRE2
represents the observed spectrum minus the quasipermanent finestructure spectrum identified in Polzin
et al. (2003). The residual (NATRE2) represents our
best estimate of the internal-wave spectrum. Two estimates
of the PATCHEX spectrum are provided: PATCHEX1
combines free-fall vertical profiler data from Gregg et al.
(1993) and long-term current meter data from Chereskin
et al. (2000), and PATCHEX2 is an estimate from a twodimensional displacement spectrum appearing in Sherman
and Pinkel (1991). Further details and a regional characterization of these data appear in PL. Finally, power
laws of a two-dimensional vertical wavenumber–frequency
spectrum, e(m, v) } v2cm2d, correspond to the power laws
of a three-dimensional vertical wavenumber–horizontal
wavenumber action spectrum n(k, m) } k2am2b with the
mapping,
a 5 c 1 2 and
b 5 d ! c.
Figure 1 suggests that the data points are not randomly distributed but have some pattern. Explaining the
location of the experimental points and making sense
out of this pattern is the main physical motivation for
this study.
c. Hamiltonian structure and wave turbulence theory
This subsection briefly summarizes the derivation in
Lvov and Tabak (2001, 2004); it is included here only
for completeness and to allow references from the core
of the paper.
The equations of motion satisfied by an incompressible stratified rotating flow in hydrostatic balance under
the Boussinesq approximation are (Cushman-Roisin
1994)
%
&
› ›z
›z
1$ "
u 5 0,
›t ›r
›r
›u
$M
1 f u? 1 u " $u 1
5 0,
›t
r0
›M
! gz 5 0.
›r
(6)
These equations result from mass conservation, horizontalmomentum conservation, and hydrostatic balance. The
2610
JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 40
M 5 P 1 grz;
f is the Coriolis parameter; and r0 is a reference density in its role as inertia, considered constant under the
Boussinesq approximation.
The potential vorticity is given by
q5
FIG. 1. The observational points. The filled circles represent the
PR spectrum [(a, b) 5 (3.5, 0.5)], the convergent numerical solution
determined in section 4c [(a, b) 5 (3.7, 0.0)], and the GM spectrum
[(a, b) 5 (4.0, 0.0)]. Circles with stars represent estimates based on
1D spectra from the western North Atlantic south of the Gulf
Stream (IWEX, FASINEX, and SFTRE/PMIII), the eastern North
Pacific (STREX/OS and PATCHEX1), the western North Atlantic
north of the Gulf Stream (Site-D), the Arctic (AIWEX), and the
eastern North Atlantic (NATRE1 and NATRE2). There are two
estimates obtained from 2D datasets from the eastern North Pacific
(SWAPP and PATCHEX2) represented as circles with crosshairs.
NATRE1 and NATRE2 represent fits to the observed spectra and
observed minus vortical mode spectra, respectively. Therefore
NATRE2 represents the best estimate of the NATRE internalwave spectrum. To conclude, 12 observational points from 10 observational programs are shown (because 2 programs produced 2
points each). Note that PATCHEX1 is indistinguishable from
SWAPP. Also note that one of the three filled circles (GM) coincides with the experimental point from Site-D.
equations are written in isopycnal coordinates with the
density r replacing the height z in its role as independent
vertical variable. Here u 5 (u, y) is the horizontal component of the velocity field; u? 5 (2y, u); $ 5 (›/›x, ›/›y)
is the gradient operator along isopycnals; M is the
Montgomery potential,
H5
ð
(
f 1 (›y/›x) ! (›u/›y)
,
P
where P 5 r/g›2M/›r2 5 r›z/›r is a normalized differential layer thickness. Because both the potential
vorticity and the fluid density are conserved along particle trajectories, an initial profile of the potential vorticity that is a function of the density will be preserved by
the flow. Hence, it is self-consistent to assume that
q(r) 5 q0 (r) 5
f
,
P0 (r)
(8)
where P0(r) 5 2g/N(r)2 is a reference stratification
profile with constant background buoyancy frequency,
N 5 (2g/(r›z/›rjbg))1/2, independent of x and y. This
assumption is not unrealistic: it represents a pancakelike distribution of potential vorticity, the result of its
comparatively faster homogenization along than across
isopycnal surfaces.
It is shown in Lvov and Tabak (2001, 2004) that the
primitive equations of motion (6) under the assumption
(8) can be written as a pair of canonical Hamiltonian
equations,
›P
dH
5!
›t
df
and
›f dH
5
,
›t
dP
(9)
where f is the isopycnal velocity potential and the
Hamiltonian is the sum of kinetic and potential energies,
'
'2
'
'
f ? !1
1
= D P(x, r) '' 1
dx dr ! [P0 1 P(x, r)]'' $f(x, r) 1
P
2
0
Here, =? 5 (2›/›y, ›/›x), D21 is the inverse Laplacian,
and r9 represents a variable of integration rather than
perturbation.
Switching to Fourier space and introducing a complex
field variable cp through the transformation
(7)
'ð
' )
g '' r
P(x, r9)''2
dr9
.
2'
r9 '
pffiffiffiffiffi
iN vp
fp 5 pffiffiffiffiffi (cp ! c!p
* ),
2gjkj
NP jkj
* ),
Pp 5 P0 ! qffiffiffi0ffiffiffiffiffiffiffi (cp 1 c!p
2gvp
(10)
(11)
DECEMBER 2010
where the frequency v satisfies the linear dispersion
relation2
ð
dpvp jcp j2 1
ð
dp012 [dp1p
(12)
1 1p2
(U p,p
1 ,p2
1 ,p2
N
1
5 pffiffiffiffiffi
(I
1 J pp ,p 1 K p,p ,p ),
1 2
1 2
4 2g kk1 k2 p,p1 ,p2
(15a)
U p,p
1 ,p2
N 1 1
5 pffiffiffiffiffi
(I
1 J !p
p1 ,p2 1 K p,p1 ,p2 ),
4 2g 3 kk1 k2 p,p1 ,p2
(15b)
rffiffiffiffiffiffiffiffiffiffi
v1 v2 2
I p,p ,p 5 !
k k1 " k2 ! [(0, 1, 2) ! (1, 2, 0)]
1 2
v
! [(0, 1, 2) ! (2, 0, 1)],
J pp
(15c)
f2 ) 2
5
p
ffi
ffi
ffi
ffiffiffiffiffiffiffiffiffiffi k k1 " k2 ! [(0, 1, 2) ! (1, 2, 0)]
1 ,p2
vv1 v2
! [(0, 1, 2) ! (2, 0, 1)]g, and
(rffiffiffiffiffiffiffiffiffiffi
v
Kp,p ,p 5 !if
(k 2 ! k22 )k1 " k?
2 1 [(0, 1, 2)
1 2
v1 v2 1
)
! (1, 2, 0)] 1 [(0, 1, 2) ! (2, 0, 1)] ,
(15d)
›
dH
c 5
,
›t p dc*p
(13)
with Hamiltonian
c*c
p *c
p *
p 1 c.c.) 1 d!p1p
This is the standard form of the Hamiltonian of a system
dominated by three-wave interactions (Zakharov et al.
1992). Calculations of interaction coefficients are tedious but straightforward tasks, completed in Lvov and
Tabak (2001, 2004). These coefficients are given by
V pp
the equations of motion (6) adopt the canonical form
i
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g2 jkj2
vp 5 f 2 1 2 2 2 ,
r0 N m
H5
2611
LVOV ET AL.
1
2
1 1p2
(V pp ,p c*c
p p cp 1 c.c.)].
1
2
1
(14)
2
a Lagrangian coordinate system is based on smallamplitude expansion to arrive at this type of equation.
In wave turbulence theory, one proposes a perturbation expansion in the amplitude of the nonlinearity,
yielding linear waves at the leading order. Wave amplitudes are modulated by the nonlinear interactions,
and the modulation is statistically described by a kinetic
equation (Zakharov et al. 1992) for the wave action np
defined by
np d(p ! p9) 5 hc*c
p p9 i.
(16)
Here h"i denotes an ensemble averaging: that is, averaging over many realizations of the random wave field.
The derivation of this kinetic equation is well studied
and understood (Zakharov et al. 1992; Lvov and Nazarenko
2004). For the three-wave Hamiltonian (14), the kinetic
equation is the one in Eq. (1), describing general internal
waves interacting in both rotating and nonrotating
environments.
The delta functions in the kinetic equation ensures
that spectral transfer happens on the resonant manifold,
which is defined as
(a)
(15e)
(c)
(
(
p 5 p1 1 p2
v 5 v1 1 v2
p2 5 p 1 p1
v2 5 v 1 v1
, (b)
(
p1 5 p2 1 p
v1 5 v2 1 v
.
,
(17)
where [(0, 1, 2) / (1, 2, 0)] and [(0, 1, 2) / (2, 0, 1)]
denote exchanges of suffixes and, for two dimensional
vector k 5 (kx, ky), k? 5 (2ky, kx).3 We stress that the
field Eq. (13) with the three-wave Hamiltonian [(12),
(14), and (15)] is equivalent to the primitive equations
of motion for internal waves (6). The approach using
Now let us assume that the wave action n is independent of the direction of the horizontal wavenumber
and is symmetric with respect to m / 2m change,
2
This dispersion relation is written in the isopycnal framework.
In the more familiar Eulerian
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the dispersion relation
qffiffiffiffiffiffiffiffiffiffiffiffiframework,
transforms into vp 5 f 2 1 (N 2 k2 /m2* ), where m*, the vertical
wavenumber in z coordinates, is given by m 5 2(g/r0N2)m.
*
3
We note that these are correct expressions, which coincide with
those given in Lvov and Tabak (2004), apart from a few ½ factors.
Note that value of the interaction matrix element is independent of horizontal azimuth because it depends only
on the magnitude of interacting wavenumbers. Therefore, one can integrate the kinetic Eq. (1) over horizontal
azimuth (Zakharov et al. 1992), yielding
np 5 n(jkj, jmj).
2612
›np
›t
5
JOURNAL OF PHYSICAL OCEANOGRAPHY
1 p2
jV pp
1 p2
j2 dm!m !m dv !v
1
2
p
p1!vp2
m91 5 m ! m92 5 m !
kk1 k2 /S01,2 .
(18)
Here, S01,2 appears as the result of integration of the
horizontal-momentum conservative delta function over
all possible orientations and is equal to the area of the
triangle with sides with the length of the horizontal
wavenumbers k 5 jkj, k1 5 jk1j and k2 5 jk2j. This is the
form of the kinetic equation that will be used to find
scale-invariant solutions in the next section.
a. Reduction of kinetic equation to the resonant
manifold
In the high-frequency limit v # f, one could conceivably neglect the effects of the rotation of the earth.
The dispersion relation (12) then becomes (Lvov and
Tabak 2001)
vp [ vk,m ’
g jkj
,
r0 N jmj
(19)
and, in this limit, the matrix element (15) retains only its
first term, I p,p ,p .
1
2
The azimuthally integrated kinetic Eq. (18) includes
integration over k1 and k2, because the integrations over
m1 and m2 can be done by using delta functions. To use
delta functions, we need to perform what is called reduction to the resonant manifold. Consider, for example, resonances of type (17a). Given k, k1, k2, and m, one
can find m1 and m2 satisfying the resonant condition by
solving simultaneous equations
m 5 m1 1 m2
and
which simplifies then to m1 of Eq. (21b).
Similarly, resonances of type (17b) yield
"
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#
m
k ! k1 ! k2 1 (k ! k1 ! k2 )2 1 4kk2
and
2k
:
m1 5 m 1 m2
8
<
m2 5 !
(22a)
"
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#
m
k 1 k1 ! k2 1 (k 1 k1 ! k2 )2 1 4kk2
m2 5 !
,
2k
:
m1 5 m 1 m2
8
<
3. Scale-invariant kinetic equation
k
k2
k
'.
5 '' 1 '' 1 ''
jmj
m1
m ! m1 '
(20)
The solutions of this quadratic equation are given by
8
<
"
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#
m
k 1 k1 1 k2 1 (k 1 k1 1 k2 )2 ! 4kk1
m1 5
and
2k
:m 5 m ! m
2
1
(21a)
8
<
"
m
k 1 k1 1 k2
2k
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #
1 (k 1 k1 1 k2 )2 ! 4kk2 ,
ð
2
(R012 ! R120 ! R201 ) dk1 dk2 dm1 dm2 ,
k
R012 5 f pp
VOLUME 40
"
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #
m
k ! k1 ! k2 ! (k ! k1 ! k2 )2 1 4kk1
m1 5
.
2k
:
m2 5 m ! m1
(21b)
Note that Eq. (21a) translates into Eq. (21b) if the indices 1 and 2 are exchanged. Indeed, exchanging indices
1 and 2 in Eq. (21a) we obtain
(22b)
and resonances of type (17c) yield
"
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#
m
k ! k1 ! k2 1 (k ! k1 ! k2 )2 1 4kk1
and
2k
:
m2 5 m 1 m1
8
<
m1 5 !
(23a)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #
m
k ! k1 1 k2 1 (k ! k1 1 k2 )2 1 4kk1
m1 5 !
.
2k
:
m2 5 m 1 m1
8
<
"
(23b)
After this reduction, a double integral over k1 and k2 is
left. The domain of integration is further restricted by
the triangle inequalities
k , k1 1 k2 , k1 , k 1 k2 ,
and
k2 , k 1 k1 .
(24)
These conditions ensure that one can construct a triangle out of the wavenumbers with lengths k, k1, and k2
and determine the domain in the (k1, k2) plane called the
kinematic box in the oceanographic literature.
Numerical evaluation of the collision integral is a
complicated but straightforward task. Interpretation
of the results, though, is more difficult, mostly because
of the complexity of the interaction matrix element
and the nontrivial nature of the resonant set. Starting
with McComas and Bretherton (1977), therefore, predictions were made based on a further simplification.
This simplification is based on the assertion that interactions between wavenumbers with extreme scale separation contribute mostly to the nonlinear dynamics.
Three main classes of such resonant triads appear, characterized by extreme scale separation. These three main
classes are as follows:
DECEMBER 2010
d
d
d
The vertical backscattering of a high-frequency wave
by a low-frequency wave of twice the vertical wavenumber into a second high-frequency wave of oppositely signed vertical wavenumber: this type of
scattering, as in Eqs. (25a), (27b), (29a), and (30a)
below, is called elastic scattering (ES).
The scattering of a high-frequency wave by a lowfrequency, small-wavenumber wave into a second,
nearly identical, high-frequency, large-wavenumber
wave: this type of scattering, as in Eqs. (25b), (27a),
(29b), and (30b) below, is called induced diffusion.
The decay of a small-wavenumber wave into two
large-vertical-wavenumber waves of approximately
one-half the frequency: this type of scattering, as in
Eqs. (26a), (26b), (28a), (28b) below, is called parametric subharmonic instability (PSI).
To see how this classification appears analytically,
we perform the limit of k1 / 0 and the limit k1 / ‘
in Eqs. (21)–(23). We will refer to the k1 or k2 / 0
limits as IR limits, whereas the k1 and k2 / ‘ limit
will be referred as a UV limit. Because the integrals
in the kinetic equation for power-law solutions will
be dominated by the scale-separated interaction, this
will help us analyze possible solutions to the kinetic
equation.
The results of the k1 / 0 limit of Eqs. (21)–(23) are
given by
*
m1 ! 2m, m2 ! !m
,
v1 $ v, v2 ; v
(25a)
*
!m1 $ m, m2 ; m
,
v1 $ v, v2 ; v
(25b)
*
m1 $ m, m2 ; !m
,
v1 ; 2v, v2 ; v
(26a)
*
!m1 $ m, m2 ; !m
,
v1 ; 2v, v2 ; v
(26b)
*
2613
LVOV ET AL.
!m1 $ m, m2 ; m
,
v1 $ v, v2 ; v
and
*
m1 ! !2m, m2 ! !m
.
v1 $ v, v2 ; v
(27a)
(27b)
We now see that the interactions (25a) and (27b) correspond to the ES mechanism, that the interactions
(25b) and (27a) correspond to the ID, and that the interactions (26a) and (26b) correspond to the PSI.
Similarly, taking the k1 and k2 / ‘ limits of Eqs. (21)–
(23), we obtain
*
m1 # m, !m2 # m
,
v1 , v2 ; v/2
(28a)
*
!m1 # m, m2 # m
,
v1 , v2 ; v/2
(28b)
*
m1 ; m/2, m2 ; !m/2
,
v1 , v2 # v
(29a)
!m2 # m
,
(29b)
m2 ; m/2
,
!m2 # m
.
*
!m1 # m,
v1 , v2 # v
*
m1 ; !m/2,
v1 , v2 # v
*
!m1 # m,
v1 , v2 # v
and
(30a)
(30b)
We now can identify the interactions (28a) and (28b) as
being PSI, the interactions (29a) and (30a) as being ES,
and the interactions (29b) and (30b) as being ID.
This classification provides an easy and intuitive tool
for describing extremely scale-separated interactions.
We will see below that one of these interactions, namely
ID, explains reasonably well the experimental data that
is available to us.
b. Convergences and divergences of the kinetic
equation
Neglecting the effects of the rotation of the earth
yields a scale-invariant system with dispersion relation
given by Eq. (19) and matrix element given only by the
Ip,p1,p2 in Eq. (15). This is the kinetic equation of Lvov
and Tabak (2001), Lvov et al. (2004), and Lvov and
Tabak (2004), describing internal waves in hydrostatic
balance in a nonrotating environment.
Proposing a self-similar separable spectrum of the
form (2), one can show from the azimuthally integrated
kinetic Eq. (18) that (Zakharov et al. 1992)
›n(ak, bm)
›n(k, m)
5 a4!2a b1!2b
›t
›t
(31)
for constants a and b. To find a steady scale-invariant
solution for all values of k and m, it is therefore sufficient
to find exponents that give a 0 collision integral for one
wavenumber. One can fix k and m, adopting for instance
k 5 m 5 1, and seek 0 of the collision integral (represented as C below) as a function of a and b,
›n(k 5 1, m 5 1)
[ C(a, b).
›t
(32)
2614
JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 40
Before embarking on numerical or analytical integration of the kinetic Eq. (18) with scale-invariant solutions (2), it is necessary to check whether the collision
integral converges. Appendix A outlines these calculations. The condition for the scale-invariant collision integral (32) to converge at the IR end, k1 or k2 / 0, is
given by
a 1 b/2 ! 7/2 , 0
and
!3 , b , 3,
(33a)
a ! 4,0
and b 5 0,
(33b)
a ! 7/2 , 0
and b 5 1,
(33c)
a 1 b ! 5 , 0 and
a ! 5 , 0 and
b . 3, or
(33d)
b , !3.
(33e)
Similarly, UV convergence as k1 and k2 / ‘ implies
that
a 1 b/2 ! 4 . 0
a ! 7/2 . 0
a ! 3.0
and
!2 , b , 2,
(34a)
and b 5 0,
(34b)
and b . 2, or
(34c)
a 1 b ! 3 . 0 and
b , !2.
(34d)
The domains of divergence and convergence are shown
in Fig. 2.
Figure 2 also displays the classes of triads dominating
the interactions. Knowing the classes of interactions that
lead to the divergences of the kinetic equation allows
us to find possible physical scenarios of the convergent
solutions or to find a possible physical regularization of
the divergences.
Note that, in addition to the two-dimensional domain
of IR convergence [the regions (33a), (33d), and (33e)],
there are two additional IR convergent line segments
given by Eqs. (33b) and (33c). These two special line
segments appear because of the b(b 2 1) prefactor to
the divergent contributions to the collision integral (A3).
Similarly, for the UV limit, in addition to the twodimensional region of convergence [(33a), (34c), and
(34d)], there is an additional special line segment of
b 5 0 [(34b)].
We see that these domains of convergence overlap
only on the segment
7/2 , a , 4 and
b 5 0.
(35)
This result contradicts the one in Lvov et al. (2004),
where a coding error led us to believe that the integrals
converge and that GM is an exact steady-state solution
to the scale-invariant kinetic equation. Note that b 5 0
FIG. 2. (a) Convergence and (b) divergence due to (a) IR
wavenumbers and (b) UV wavenumbers. The integral converge for
the exponents in the shaded regions or on the segments (b 5 0, 7/ 2 ,
a , 4) and (b 5 1, 3 , a , 7/ 2). Dashed lines distinguish the domains
where the indicated named triads dominate the singularity.
corresponds to wave action independent of vertical
wavenumbers, ›n/›m 5 0. Existence of the b 5 0 line
will allow us to find novel convergent solution in section
4. We also note that the IR segment on b 5 0 coincides
with one of the ID solutions determined in section 6. The
other segment on b 5 1 does not coincide with the ID
solution in section 6, because the scale-invariant system
has higher symmetry than the system with Coriolis effect.
4. A novel convergent solution
To find out whether there is a steady solution of the
kinetic equation along the convergent segment (35), we
substitute the power-law ansatz (2) with b 5 0 into the
azimuthally integrated kinetic Eq. (18). We then compute numerically the collision integral as a function of
a for b 5 0. To this end, we fix k 5 m 5 1 and perform
a numerical integration over the kinematic box (24),
DECEMBER 2010
2615
LVOV ET AL.
FIG. 3. Value of the collision integral as a function of a on the
convergent segment, b 5 0.
reducing the integral to the resonant manifold as described in section 3a.
The result of this numerical integration is shown in
Fig. 3. The figure clearly shows the existence of a steady
solution of the kinetic Eq. (18) near a ffi 3.7 and b 5 0.
This is, therefore, the only convergent steady solution
to the scale-invariant kinetic equation for the internalwave field. It is highly suggestive that it should exist so
close to the GM spectrum, a 5 4 and b 5 0 for large
wavenumbers, the most agreed upon fit to the spectra
observed throughout the ocean. It remains to be seen
whether and how this solution is modified by inclusion of
background rotation.
FIG. 4. Signs of the divergences where the both IR and UV
contributions diverge. The left symbols show the signs due to IR
wavenumbers and the right symbols show the signs due to UV
wavenumbers.
Similarly, at the UV end, resonant interactions are
dominated by the ID singularity for 22 , b , 2 (Fig. 2).
The sign of the divergence is given by b [Eq. (A4)]. At
the UV end for b . 2, where ES determines the divergences, the sign of the singularity is given by 2b: that
is, the sign is negative [Eq. (A5)]. Figure 4 shows the
signs of the divergences where both the IR and UV
contributions diverge: the left sign corresponds to the
IR contribution, and the right sign corresponds to the
UV contribution.
Hence, in the regions,
7 , 2a 1 b , 8 and
5. Balance between divergences
The fact that the collision integral C diverges for almost all values of a and b can be viewed both as a challenge and as a blessing. On one hand, it makes the
prediction of steady spectral slopes more difficult, because it now depends on the details of the truncation of
the domain of the integration. Fortunately, it provides
a powerful tool for quantifying the effects of fundamental players in ocean dynamics, most of which live on
the fringes of the inertial subrange of the internal-wave
field: the Coriolis effect, as well as tides and storms, at
the IR end of the spectrum and wave breaking and
dissipation at the UV end. The sensitive response of the
inertial subrange to the detailed modeling of these scaleseparated mechanisms permits, in principle, building
simple models in which these are the only players, bypassing the need to consider the long range of wave scales
in between.
At the IR end, the resonant interactions are dominated by the ID singularity for 23 , b , 3 (Fig. 2). The
sign of the divergences is given by 2b(b 2 1) [Eq. (A3)].
7 , 2a 1 b,
!2 , b , 1 or
a , 3, and b . 2,
(36a)
(36b)
the divergences of the collision integral at the IR and
UV ends have opposite signs. Then formal solutions can
be constructed by having these two divergences cancel
each other out.
This observation justifies the existence of the PR solution (5). Indeed, the PR spectrum has divergent powerlaw exponents at the both ends. One can prove that the
PR spectrum is an exact steady solution of the kinetic
Eq. (1) by applying the Zakharov–Kuznetsov conformal
mapping for systems with cylindrical symmetry (Zakharov
1968, 1967; Kuznetsov 1972). This Zakharov–Kuznetsov
conformal mapping effectively establishes a map between
the neighborhoods of zero and infinity. Making these
two contributions cancel pointwise yields the solution (5).
For this transformation to be mathematically applicable,
the integrals have to converge. This transformation leads
only to a formal solution for divergent integrals. Some
other transformation, as we explain below, may lead to
completely different solutions.
2616
JOURNAL OF PHYSICAL OCEANOGRAPHY
The PR spectrum was first found by Pelinovsky and
Raevsky (1977). However, they realized that it was
only a formal solution. The solution was found again
in Caillol and Zeitlin (2000) through a renormalization
argument and in Lvov and Tabak (2001, 2004) within an
isopycnal formulation of the wave field.
The idea of a formal solution, such as PR, can be
generalized quite widely: in fact, any point in the regions
with opposite-signed divergences can be made into a
steady solution under a suitable conformal mapping that
makes the divergences at zero and infinity cancel each
other, as does the Zakharov–Kuznetsov transformation.
Such generalized Zakharov–Kuznetsov transformation
is an extension of the idea of principal value for a divergent integral, whereby two divergent contributions
are made to cancel each other through a specific relation between their respective contributions.
Indeed, in the ocean internal waves cannot have zero
horizontal wavenumber. Rather, a smallest horizontal
wavenumber exists, which corresponds to largest horizontal scales of internal waves. The largest wavenumber
that was observed is on the order of thousands of kilometers. On larger scales, b effects become important,
and they prevent wavenumbers to achieve even smaller
scales. Other effects possibly affecting small wavenumbers
include ocean storms, interactions with large scale vortices and shear, and ocean boundaries. Similarly, there is
no infinitely large wavenumber for internal waves, but
rather there is possibly large horizontal wavenumber that
is affected by wave breaking, interaction with the turbulence, and other processes.
The idea of a generalized Zakharov–Kuznetsov transformation leading to an infinite number of steady states
provides a possible explanation for the variability of the
power-law exponents of the quasi-steady spectra: inertial subrange spectral variability is to be expected when
it is driven by the nonlocal interactions. The natural local
variability of players outside the inertial range translates
into a certain degree of nonuniversality. Such players
include storms and tides, as well as possible geometrical
constrains and interactions with large-scale shear and
vortices. Investigation of the nature of possible balances
between IR and UV divergences is outside of scope of
the present paper.
6. Regularization by the Coriolis effect
Physically, the ocean does not perform generalized
Zakharov–Kuznetsov transformation. However, in the
ocean there are finite boundaries in the frequency domain. In particular, the inertial frequency f provides
a truncation for the IR part of the spectrum, whereas the
VOLUME 40
UV truncation is provided by the buoyancy frequency N.
These two frequencies vary from place to place, giving
grounds for spectral variability. Consequently, the integrals are not truly divergent, but rather they have
a large numerical value dictated by the location of the
IR cutoff.
Note that the f / 0 calculations presented in this
section describe intermediate frequencies v such that
v # f . 0. Consequently, these intermediate frequencies feel inertial frequency as being small. These calculations will not be applicable for equatorial regions,
where f is identically zero.
Observe that all the experimental points are located
in regions of the (a, b) domain a . (2b 1 7)/2 for which
the integral diverges in the IR region. Also note that 5
out of 12 experimental points are located in the region
where collision integrals are UV convergent: that is, in
a . 2b/2 1 4. The UV region is therefore assumed to
be either subdominant or convergent in this section,
where we study the regularization resulting from a finite
value of f.
We note that, because we consider scale-invariant
case only in the present paper, the IR cutoff can equivalently be considered as k approaching smallest possible
value or equivalently when v approaches its smallest
value. Because the IR cutoff is given by f, a frequency,
it is easier to analyze the resulting integral in (v1, m1)
rather than in the more traditional (k1, m1) domain
utilized in Müller et al. (1986) and the previous sections.
We emphasize that the present paper concerns itself
with scale-invariant wave action spectrum (2). The scaleinvariant wave action can be translated from (k1, m1) to
(v1, m1) domain with out loss of generality. This statement would not be true for the realistic Garrett–Munk
spectrum or any other oceanic spectrum because of the
non-scale-invariant form of the linear dispersion relation
(12). Such spectra will be analyzed in subsequent publication (LPY).
Therefore, to proceed, we assume a power-law spectrum, similar to Eq. (2), but in the (v, m) space,
~
nv,m } va~mb .
(37)
We need to transform the wave action as a function of k
and m to a function of v and m. This is done in appendix
B. The relation between a, b and a~, b~ reads as
a~ 5 !a, b~ 5 !a ! b.
Thus, we need to express both the kinetic equation
and the kinematic box in terms of frequency and vertical
wavenumber. For this, we use the dispersion relation
DECEMBER 2010
2617
LVOV ET AL.
(12) to express k in terms of v in the description of the
kinematic box (24),
v1 , E3 (m1 ),
if
v1 , E4 (m1 ),
v1 , E1 (m1 )
(38b)
v1 . E2 (m1 ),
v1 . E4 (m1 )
m1 . 0, v1 , v; and
v1 . E3 (m1 ),
if
(38a)
m1 , 0, v1 , v;
v1 , E2 (m1 ),
if
v1 . E1 (m1 )
m1 , 0, v1 . v;
v1 . E3 (m1 ),
if
v1 . E4 (m1 ),
v1 , E1 (m1 ),
(38c)
v1 , E2 (m1 )
m1 . 0, v1 . v,
(38d)
FIG. 5. The kinematic box in the (v1, m1) domain. Two disconnected regions where v1 , v depict regions with ‘‘sum’’ interactions:
namely, v 5 v1 1 v2 and m 5 m1 1 m2 types of the resonances. The
connected regions where v1 . v depict ‘‘difference’’ resonances,
v2 5 v1 2 v and m2 5 m1 2 m. The parameters are chosen so that
f/v 5 0.1. Frequency f is marked on the top of the graph, and frequency N is outside of the region of frequencies shown on the
graph.
where we have introduced the four curves in the (v1, m1)
domain that parameterize the kinematic box,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
! !f 2 1 v2 1 ! f 2 1 (v ! v1 )2
E1 (v1 ) 5 m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,
!f 2 1 v21 1 !f 2 1 (v ! v1 )2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
! f 2 1 v2 1 !f 2 1 (v ! v1 )2
E2 (v1 ) 5 m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,
! f 2 1 v21 1 !f 2 1 (v ! v1 )2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
! !f 2 1 v2 1 ! f 2 1 (v ! v1 )2
E3 (v1 ) 5 m qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , and
! !f 2 1 v21 1 ! f 2 1 (v ! v1 )2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!f 2 1 v2 1 !f 2 1 (v ! v1 )2
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .
E4 (v1 ) 5 m
! !f 2 1 v21 1 ! f 2 1 (v ! v1 )2
d
d
d
d
d
d
d
d
d
The kinematic box in the (v, m) domain is shown in
Fig. 5. Note that the region (38b) and (38c) can be
transferred to each other by interchanging indices 1 and 2;
consequently, two disconnected v1 , v regions look like
mirrored and shifted copies of each other. To help in the
transition from the traditional kinematic box (24) to the
kinematic box in (v, m) domain, the following limits were
identified:
d
d
d
An advantage of the (v, m) presentation for the kinematic box is that it allows a transparent reduction to
the resonant manifold. A disadvantage is the curvilinear boundaries of the box, which requires more sophisticated analytical treatment.
Equation (18) transforms into
ID1 is the ID limit of Eq. (25b) with indices 1 and 2
being flipped;
›
1
n[k(v, m), m] 5
›t
k
ð
dv1 dm1 J
' 0 '2
'V '
12
S01,2
ID2 is the ID limit of Eq. (27a) with indices 1 and 2
being flipped;
ID3 is the ID limit of Eq. (25b);
ID4 is the ID limit of Eqs. (29b) and (30b);
PSI1 is the PSI limit of Eq. (28a);
PSI2 is the PSI limit of Eq. (28b);
PSI3 is the PSI limit of Eq. (26a);
PSI4 is the PSI limit of Eq. (26b);
ES1 is the ES limit of Eq. (25a);
ES2 is the ES limit of Eq. (27b) with indices 1 and 2
being flipped;
ES3 is the ES limit of Eq. (27b) with indices 1 and 2
being flipped; and
ES4 is the ES limit of Eq. (29a).
'
[n1 n2 ! n(n1 1 n2 )]'v
3 [nn2 ! n1(n 1 n2 )]jv 5v !v,m 5m !m .
2
1
2
1
' 1 '2
ð
'V '
2
dv1 dm1 J 02
!
5v!v
,
m
5m!m
2
1
2
1
k
S12,0
(39)
2618
JOURNAL OF PHYSICAL OCEANOGRAPHY
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
We have used the dispersion relation ki 5 mi v2i ! f 2
and defined J as the Jacobian of the transformation from
(k1, k2) into (v1, v2) times the kk1k2 factor,
J 5 kk1 k2
1) ID1 region: In this region, m1 is slightly bigger than
m, v1 is slightly smaller than v, and v2 and m2 are
both very small. This region can be obtained from the
region (38b) above by interchanging indices 1 and 2.
In this region, the wave action n2 is much smaller
than wave action n and n1:
s
f
1 (v 1 )
E 3 (v 1 )
3) ID3 region: In this region, v1 is small and m1 is small
and negative. This corresponds to the region (38b),
where wave action obeys
n1 # n, n2 .
Note that this region can be obtained from the region
ID1 by flipping indices 1 and 2. Consequently, only
one of the ID1 and ID3 should be taken into account,
with a factor of 2 multiplying the respective contribution. This also can be seen from Fig. 5 as ID1 and
ID3 regions are shifted mirror images of each other.
Making these simplifications and taking into account
the areas of integration in the kinematic box, we obtain
n2 # n, n1 .
ð f 1v ð ð E
2) ID2 region: In this region, v1 is slightly bigger than
v 1 f, where v2 5 v1 2 v is small and m2 5 m1 2 m is
negative and small. This is the region (38d). Also,
n2 # n, n1 .
dk1 dk2
.
dv1 dv2
In Fig. 5, there are three ID regions (or corners) with
significant contribution to the collision integral in the IR
limit:
›
2
n[k(v, m), m] 5
›t
k
dm1 J
' 0 '2
'V '
1,2
S01,2
2
n1 (n2 ! n) dv1 !
k
where the small parameter vs is introduced to restrict
the integration to a neighborhood of the ID corners. The
arbitrariness of the small parameter will not affect the
end result below.
To quantify the contribution of near-inertial waves to
a (v, m) mode, we write
! ; f $ v 5 1.
Subsequently, near the region ID3 of the kinematic box,
we write
The integral over ! diverges at ! 5 0, if f 5 0 and if
6 1 a~ 1 b~ . !1.4
However, if we postpone taking f 5 0 limit, we see that
the integral is 0 to leading order if
4
or a~ ! b~ 5 0
Naturally, this condition coincides with (33a).
(42)
ð v! f
v! f !vs
dv1
ðE
1 (v1 )
E3 (v1 )
dm1 J
' 2 '2
'V '
1,0
S21,0
n2 (n ! n1 )
(40)
v1 5 f 1 !,
whereas near the ID2 corners of the kinematic box we
write
v1 5 v 1 f 1 !.
We then expand the resulting analytical expression (40)
in powers of ! and f without making any assumptions
on their relative sizes. These calculations, including the
integration over vertical wavenumbers m1, are presented
in appendix C. The resulting expression for the kinetic
equation is given by
›
p
~ a ! 3(3 1 b)]
~ 3 m512b~v!31~a!b~
n[k (v, m), m] 5
(~
a ! b)[~
›t
4k
~ 50
a~ ! 3(3 1 b)
VOLUME 40
ðm
~
d!(! 1 f )41~a1b (!2 1 2! f 1 17 f 2 ).
(41)
0
or, in terms of a and b,
9 ! 2a ! 3b 5 0
or b 5 0.
(43)
This is the family of power-law steady-state solutions
to the kinetic equations dominated by infrared ID interactions. These steady states are identical to the ID
stationary states identified by McComas and Bretherton
(1977), who derived a diffusive approximation to their
collision integral in the infrared ID limit. Note that
DECEMBER 2010
LVOV ET AL.
McComas and Müller (1981) interpreted b 5 0 as a
no-action flux in vertical wavenumber domain, whereas
9 2 2a 2 3b 5 0 is a constant action flux solution.
We note that one can use the eikonal approach to
describe these types of interactions (Flatté et al. 1985;
Broutman et al. 2004; Müller et al. 1986; Henyey et al.
1986). An advantage of the eikonal approach versus our
scale-invariant analysis is that it allows us to consider
not only scale-invariant interactions in separable spectrum. The possible disadvantage of the eikonal approach is that construction of a transport theory is far
less rigorous. For a more detailed discussion on the differences between the resonant interaction approximation
and the eikonal approach, we refer the reader to Polzin
(2004).
What is presented in this section is a rigorous asymptotic derivation of Eq. (43). These ID solutions help
us to interpret observational data of Fig. 1 that are currently available to us. The value added associated with
a rigorous asymptotic derivation is the demonstration
that the ID stationary states are meaningful only in the
IR divergent part of the (a, b) domain.
7. Conclusions
The results in this paper provide an interpretation
of the variability in the observed spectral power laws.
Combining Figs. 1, 2, and 4 with Eqs. (4), (5), and (43)
produces the results shown in Fig. 6.
A nonrotating scale-invariant analysis provides two
subdomains in which the kinetic equation converges in
either the UV or the IR limit (light gray shading), two
subdomains in which the kinetic equation diverges in
both UV and IR limits but with oppositely signed values
(dark gray shading) and a single domain with similarly
signed UV and IR divergences (black shading). In this
nonrotating analysis, a stationary state is possible only
for oppositely signed divergences: that is, within the
dark gray shaded regions. Six of the observational points
lie in IR and UV divergent subdomains, and a seventh
(Site-D) is on the boundary with the IR divergent–UV
convergent subdomain. Two of the observational points
lie in the domain of IR and UV divergence having similar signs, which does not represent a possible solution
in the nonrotating analysis. These two points lie close to
the boundaries of the ‘‘forbidden’’ (black shaded) region, and subtracting the vortical contribution from one
(NATRE1) returns a ‘‘best’’ estimate of the internalwave spectrum (NATRE2) that lies outside of the forbidden region.
All the data lie in an IR divergent regime and hence
a regularization of the kinetic equation is performed by
including a finite lower frequency of f. This produces
2619
FIG. 6. The observational points and the theories: The filled
circles represent the PR spectrum, the convergent numerical solution determined in section 4c, and the GM spectrum. Circles with
stars represent power-law estimates based on 1D spectra. Circles
with crosshairs represent estimates based on 2D datasets. See Fig. 1
for the identification of the field programs. Light gray shading
represents regions of the power-law domain for which the collision
integral converges in either the IR or UV limit. The dark gray
shading represents the region of the power-law domain for which
the IR and UV limits diverge and have opposite signs. The region
of black shading represents the subdomain for which both the IR
and UV divergences have the same sign: that is, when large contributions from interactions with very small and very large wavenumbers have the same sign. Overlaid as solid white lines are the
ID stationary states.
a family of stationary states, the induced diffusion
stationary states. These stationary states collapse much
of the observed variability. The exception is the NATRE
spectrum.
Summarizing the paper, we have analyzed the scaleinvariant kinetic equation for internal gravity waves and
shown that its collision integral diverges for almost all
spectral exponents. Figure 6 shows that the integral
nearly always diverges at zero, at infinity, or at both
ends. This means that, in the wave turbulence kinetic
equation framework, the energy transfer is dominated by
the scale-separated interactions with large and/or small
scales.
The only exception where the integral converges is
a segment of a line, 7/ 2 , a , 4, with b 5 0. On this
convergent segment, we found a special solution, (a, b) 5
(3.7, 0). This new solution is not far from the largewavenumber asymptotic form of the Garrett–Munk spectrum, (a, b) 5 (4, 0).
We have argued that there are two subdomains of
power-law exponents that can yield quasi-steady solutions of the kinetic equation. For these ranges of exponents, the contribution of the scale-separated interactions
2620
JOURNAL OF PHYSICAL OCEANOGRAPHY
due to the IR and UV wavenumbers can be made to
approximately balance each other. The Pelinovsky–
Raevsky spectrum is a special case of this scenario.
The scenario, in which the energy spectrum in the
inertial subrange is determined by the nonlocal interactions, provides an explanation for the variability of the
power-law exponents of the observed spectra. They are
a reflection of the variability of dominant players outside of the inertial range, such as the Coriolis effect,
tides, and storms.
This possibility was further investigated by introducing rotation and then pursuing a rigorous asymptotic
expansion of the kinetic equation. In doing so, we obtain
the induced diffusion stationary states that appear as
white lines in Fig. 6, which had previously been determined through a diffusive approximation. Much of the
observed oceanic variability lies about these stationary
states in the IR divergent subdomain.
A more detailed review of available observational
data used for this study appears in PL. Numerical evaluation of the complete non-scale-invariant kinetic equation of the Garrett–Munk spectrum is presented in LPY
in which we also consider waves that are slightly off resonant interactions. The theory, experimental data, and
results of numerical simulations in Lvov and Yokoyama
(2009) all hint at the importance of the IR contribution
VOLUME 40
to the collision integral. The nonlocal interactions with
large scales will therefore play a dominant role in forming the internal-wave spectrum. To the degree that the
large scales are location dependent and not universal, the
high-frequency, high-vertical-wavenumber internal-wave
spectrum ought to be affected by this variability. Consequently, the internal-wave spectrum should be strongly
dependent on the regional characteristics of the ocean,
such as the local value of the Coriolis parameter and
specific features of the spectrum, specifically for nearinertial frequencies.
Acknowledgments. This research is supported by NSF
CMG Grants 0417724, 0417732 and 0417466. YL is also
supported by NSF DMS Grant 0807871 and ONR Award
N00014-09-1-0515. We are grateful to YITP in Kyoto
University for allowing us to use their facility.
APPENDIX A
Asymptotics of Collision Integral in Infrared
and Ultraviolet Limits
Let us integrate Eq. (18) over m1 and m2,
ð
.' '
1
' 0
(T 01,2 ! T 12,0 ! T 20,1 ) dk1 dk2 , T 01,2 5 kk1 k2 jV pp p j2 f pp p ('g09
1,2 S1,2 ),
1 2
1 2
k
›t
'
dg01,2 (m1 )''
k
k2
k
09
'.
g1,2 (k1 , k2 ) 5
, g01,2 (m1 ) 5 ! '' 1 '' ! ''
'
'
m
dm1
m1
m ! m1 '
m1 5m*
(
k
,
k
)
1
1 2
›np
5
*
Here, g09
1,2 appears because of dvp!vp1!vp2, and m1 (k1, k2)
is given by the resonant conditions (21)–(23).
a. Infrared asymptotics
We consider the asymptotics of the integral in Eq.
(A1) as k1 / 0. We employ the independent variables x
and y, where k1 5 kx, k2 5 k(1 1 y), x, y 5 O(!), x . 0,
and 2x , y , x. In this limit of ! / 0, n1 # n, n2. In this
limit, Eqs. (21a) and (23b), Eqs. (21b) and (23a), and
Eqs. (22a) and (22b) correspond to ES [Eqs. (25a) and
(27b)], ID [Eqs. (25b) and (27a)], and PSI [Eqs. (25a)
and (27b)], respectively. Without loss of generality, m is
set to be positive.
Assuming the power-law spectrum of the wave action,
n(k, m) 5 jkj2ajmj2b, we make Taylor expansion for the
integrand of the kinetic Eq. (A1) as powers of !: that is, x
and y. Then, we get Table A1, which shows the leading
orders of the each terms according to the asymptotics. The
(A1)
leading order of the collision integral is given by ID when
23 , b , 3. Therefore, we are going to show the procedure to get the leading order of ID (21b) and (23a) below.
TABLE A1. Asymptotics as k1 / 0. ES [(21a) and (23b)] gives
!2a15 because of the symmetry of y. ID [(21b) and (23a)] gives
!2a2(b27)/2 (!2a14) because of the second cancellation. PSI [(22a)
and (22b)] gives !2a2b15. The asymptotics for b 5 0 appear in
parentheses.
p
p
Eq.
m1
m2
v1
v2
V pij pk
f pij pk
gi9j,k
T ij,k
(21a)
(21b)
!0
!1/2
!0
!0
!1
!1/2
!0
!0
!1/2
!1/4
!0
!0
(22a)
(22b)
(23a)
!1
!1
!1/2
!0
!0
!0
!0
!0
!1/2
!0
!0
!0
!1
!1
!1/4
(23b)
!0
!0
!1
!0
!1/2
!2a11
!2a2(b21)/2
(!2a11)
!2a2b
!2a2b
!2a2(b21)/2
(!2a11)
!2a11
!2a12
!2a2b//211
(!2a13/2)
!2a2b13
!2a2b13
!2a2b//211
(!2a13/2)
!2a12
!21
!21
!0
!0
DECEMBER 2010
2621
LVOV ET AL.
As ! / 0, n2 / n for ID solutions. Therefore, the
leading order of f 01,2 ; n1 (n2 ! n) and f 20,1 ; n1 (n ! n2 )
is O(!2a2(b21)/2). The order O(!2a2b/2) is canceled as
! / 0. This is called the first cancellation. It must be
noted that the leading order when b 5 0 is ½ larger than
that when b 6¼ 0 because ›n/›m 5 0. The leading orders
when b 5 0 are written in parentheses in Table A1.
The leading order of the integrand in Eq. (A1) is
written as
x!a!(b11)/2 y
T 01,2 ! T 12,0 ! T 20,1 } k!2a13 m!2b11 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(x 1 y)(x ! y)
3 f!2ay2 ! b[(1 ! b)y(x 1 y)
! 2x(x ! y)] 1 b(b 1 1)xyg.
(A2)
Therefore, the integrand has O(!2a2(b25)/2). The term
that has O(!2a2b/212) is canceled because T 20,1 ! T 01,2
(and T 12,0 ! 0) as ! / 0. This is the second cancellation.
Finally, we get the leading order of the kinetic equation after integration over y from 2x to x,
ð
›np
} !b(1 ! b)k4!2a m1!2b x!a!(b!5)/2 dx. (A3)
›t
0
The integral has O(!2a2(b27)/2). Consequently, the integral converges if
a 1 (b ! 7)/2 , 0 and
),
The integral for the PR spectrum, which gives O(!
diverges as k1 / 0. However, the integral for the GM
spectrum converges because b 5 0 and the next order
is O(1).
It should be noted that the leading order when b 5 1
is ½ larger than that when b 6¼ 0, 1 because a balance
between first- and second-order derivative is made. The
leading order when b 5 1 is O(!2a17/2). It is also helpful
to note that T 01,2 ! T 12,0 ! T 21,0 5 O(«!a12 ) for ES because of no second cancellation. However, the collision
integral has O(!2a15) because of symmetry of y. Therefore, the integral which is dominated by ES converges
b , !3.
Similarly, the integral that is dominated by PSI converges,
p
p
Eq.
m1
m2
v1
v2
V pij pk
f pij pk
gi9j,k
T ij,k
(21a)
(21b)
(22a)
(22b)
!21
!21
!0
!21/2
!21
!21
!0
!21/2
!0
!0
!21
!21/2
!0
!0
!21
!21/2
!0
!0
!21
!21
!1
!1
!21
!1/2
(23a) !0
!0
!21 !21
21/2
21/2
(23b) !
!
!21/2 !21/2
!21
!21
!a1b
!a1b
!a
!a1b/211
(!a13/2)
!a
!a1b/211
(!a13/2)
!a1b22
!a1b22
!a22
!a1b/223
(!a25/2)
!a22
!a1b/223
(!a25/2)
!21
!1/2
variables x and y as k1 5 k/2[1 1 (1/x) 1 y] and k2 5
k/2[1 1 (1/x) 2 y], where x 5 O(!) and 21 , y , 1. Again,
m . 0 is assumed.
The leading orders are obtained by the similar manner
used in the IR asymptotic and are summarized in Table A2.
The leading order of the integral is given by ID, whose
wavenumbers are given by Eqs. (22b) and (23b), when
22 , b , 2. In this limit, no second cancellation is made.
As the result of the perturbation theory, we get the
leading order,
ð
›np
} k4!2a m1!2b b xa1b/2!5 dx.
(A4)
›t
0
It has O(!a1b/224). Therefore, the integral converges if
a 1 b/2 ! 4 . 0
!3b3.
21/4
a ! 5 , 0 and
TABLE A2. Asymptotics as k1 / ‘. PSI [(21a) and (21b)] gives
!a1b23. ES [(22a) and (23a)] gives !a23. ID [(22b) and (23b)] gives
!a1b/224 (!a27/2). The asymptotics for b 5 0 appear in parentheses.
and
!2 , b , 2.
The integral for the PR spectrum, which gives O(!21/4),
diverges as k1 / ‘. The integral for the Garrett–Munk
spectrum, which gives O(!0), converges because b 5 0.
Similarly,
ð
›np
(A5)
} !k4!2a m1!2b b xa!2 dx
›t
0
for ES, which is dominant for b . 2. Consequently, the
integral converges also if
a ! 3.0
and b . 2.
In the same manner, the convergent domain of the integral for PSI is given by
a1b ! 3.0
and b , !2.
APPENDIX B
a1b ! 5,0
and b . 3.
b. Ultraviolet asymptotics
Next, we consider the limit k1 / ‘. In this case, k2
also approaches infinity. We employ the independent
Frequency–Vertical-Wavenumber and
Horizontal–Vertical-Wavenumber Spectrum
The theoretical work presented below addresses the
asymptotic power laws of a three-dimensional action
2622
JOURNAL OF PHYSICAL OCEANOGRAPHY
spectrum. To connect with that work, note that a horizontally isotropic power-law form of the three-dimensional
wave action n(k, m) is given by Eq. (2).
The corresponding vertical wavenumber–frequency
spectrum of energy is obtained by transforming nk,m from
wavenumber space (k, m) to the vertical wavenumber–
frequency space (v, m) and multiplying by frequency. In
the high-frequency, large-wavenumber limit,
E(m, v) } v2!a m2!a!b .
The total energy density of the wave field is then
E5
ð
v(k, m)n(k, m) dk dm 5
ð
E(v, m) dv dm.
Thus, we also find it convenient to work with the wave
action spectrum expressed as a function of v and m
Therefore, we also introduced (37). The relation between a, b and a~, b~ reads as
b~ 5 !a ! b.
a~ 5 !a,
Before proceeding, note the following symmetries:
E1 (v1 5 v ! v91 ) 5 m ! E2 (v91 ),
E3 (v1 5 v ! v91 ) 5 m ! E3 (v91 ),
These symmetries explain why two disconnected regions
on Fig. 5 look like mirrored and shifted copies of each
other. These symmetries further allow us simplification
of evaluation of ID contribution by noticing that the
contribution from ID1 is equal to the contribution from
ID3. To quantify the contribution of near-inertial waves
to a (v, m) mode, we write
! ; f $ v 5 1.
Subsequently, in the domain ID3 we write
v1 5 f 1 !,
in P 1 and in the domain ID2 we write
v1 5 v 1 f 1 !
Asymptotic Expansion for Small f Values
In this section, we perform the small f calculations of
section 6. We start from the kinetic equation written as
Eq. (40). There we change variables in the first line of
Eq. (40) as
m1 5 E3 (v1 ) 1 m[E1 (v1 ) ! E3 (v1 )]
2
!
k
f
ð v! f
v! f !vs
ð1
0
dmP 1
dv1
ð1
0
dmP 2 .
(C1)
Here, we introduced integrands P 1 and P 2 to be
P1 5 J
' 0 '2
'V '
n1 (n2 ! n)[E1 (v1 ) ! E3 (v1 )]
P2 5 J
S01,2
n2 (n ! n1 )[E2 (v1 ) ! E3 (v1 )].
1,2
S01,2
' 0 '2
'V '
1,2
f . 0, and
0 , m , 1.
Define
and
This allows us to expand P1, P2, P3, and P4 in powers of
f and !. We perform these calculations analytically on
Mathematica software. Mathematica was then able to
perform the integrals of P 1 and P 2 over m from 0 to 1 in
Eq. (C1) analytically. The result is given by Eq. (41).
Then, Eq. (40) becomes the following form:
dv1
m . 0, ! . 0,
P 2 5 P3 1 P4 .
m1 5 E3 (v1 ) 1 m[E2 (v1 ) ! E3 (v1 )].
s
in P 2 . Furthermore, we expand P 1 and P 2 in powers of !
and f without making any assumptions of the relative
smallness of f and !. We use the facts that
P 1 5 P1 1 P2
and in the second line of Eq. (40) as
ð f 1v
and
E4 (v1 5 v ! v91 ) 5 m ! E4 (v91 ).
APPENDIX C
›
2
n[k(v, m), m] 5
›t
k
VOLUME 40
and
(C2)
REFERENCES
Anderson, S., 1992: Shear, strain, and thermohaline vertical fine
structure in the upper ocean. Ph.D. thesis, University of California, San Diego, 155 pp.
Broutman, D., J. W. Rottman, and S. D. Eckermann, 2004: Ray
methods for internal waves in the atmosphere and ocean.
Annu. Rev. Fluid Mech., 36, 233–253.
Caillol, P., and V. Zeitlin, 2000: Kinetic equations and stationary
energy spectra of weakly nonlinear internal gravity waves.
Dyn. Atmos. Oceans, 32, 81–112.
Cairns, J. L., and G. O. Williams, 1976: Internal wave observations
from a midwater float, 2. J. Geophys. Res., 81, 1943–1950.
DECEMBER 2010
LVOV ET AL.
Chereskin, T. K., M. Y. Mooris, P. P. Niiler, P. M. Kosro, R. L. Smith,
S. R. Ramp, C. A. Collins, and D. L. Musgrave, 2000: Spatial
and temporal characteristics of the mesoscale circulation
of the California Current from eddy-resolving moored
and shipboard measurements. J. Geophys. Res., 105, 1245–
1269.
Cushman-Roisin, B., 1994: Introduction to Geophysical Fluid Dynamics. Prentice-Hall, 320 pp.
D’Asaro, E. A., 1984: Wind forced internal waves in the North
Pacific and Sargasso Sea. J. Phys. Oceanogr., 14, 781–794.
——, 1995: A collection of papers on the ocean storms experiment.
J. Phys. Oceanogr., 25, 2817–2818.
——, and M. D. Morehead, 1991: Internal waves and velocity fine
structure in the Arctic Ocean. J. Geophys. Res., 96, 12 725–
12 738.
Eriksen, C., D. Rudnick, R. Weller, R. T. Pollard, and L. Regier,
1991: Ocean frontal variability in the Frontal Air-Sea Interaction Experiment. J. Geophys. Res., 96, 8569–8591.
Flatté, S. M., F. S. Henyey, and J. A. Wright, 1985: Eikonal calculations of short-wavelength internal-wave spectra. J. Geophys. Res., 90, 7265–7272.
Foffonoff, N. P., 1969: Spectral characteristics of internal waves in
the ocean. Deep-Sea Res., 16, 58–71.
Garrett, C. J. R., and W. H. Munk, 1972: Space-time scales of internal waves. Geophys. Fluid Dyn., 3, 225–264.
——, and ——, 1975: Space-time scales of internal waves: A progress report. J. Geophys. Res., 80, 281–297.
——, and ——, 1979: Internal waves in the ocean. Annu. Rev. Fluid
Mech., 11, 339–369.
Gregg, M. C., H. E. Seim, and D. B. Percival, 1993: Statistics of
shear and turbulent dissipation profiles in random internal
wave fields. J. Phys. Oceanogr., 23, 1777–1799.
Hasselmann, K., 1966: Feynman diagrams and interaction rules of
wave-wave scattering processes. Rev. Geophys., 4, 1–32.
Henyey, F. S., J. Wright, and S. M. Flatté, 1986: Energy and action
flow through the internal wave field: An eikonal approach.
J. Geophys. Res., 91, 8487–8495.
Keffer, T., 1983: The baroclinic stability of the Atlantic North
Equatorial Current. J. Phys. Oceanogr., 13, 624–631.
Kenyon, K. E., 1966: Wave-wave scattering for gravity waves and
Rossby waves. Ph.D. thesis, University of California, San Diego,
93 pp.
——, 1968: Wave-wave interactions of surface and internal waves.
J. Mar. Res., 26, 208–231.
Kuznetsov, E. A., 1972: On turbulence of ion sound in plasma in
a magnetic field. Zh. Eksper. Teoret. Fiz, 62, 584–592.
Levine, M. D., J. D. Irish, T. E. Ewart, and S. A. Reynolds, 1986:
Simultaneous spatial and temporal measurements of the internal wavefield during MATE. J. Geophys. Res., 91, 9709–
9719.
——, C. A. Paulson, and J. H. Morison, 1987: Observations of internal gravity waves under the arctic pack ice. J. Geophys.
Res., 92, 779–782.
Lvov, Y. V., and E. G. Tabak, 2001: Hamiltonian formalism and the
Garrett–Munk spectrum of internal waves in the ocean. Phys.
Rev. Lett., 87, 168501, doi:10.1103/PhysRevLett.87.168501.
——, and S. Nazarenko, 2004: Noisy spectra, long correlations, and
intermittency in wave turbulence. Phys. Rev., 69, 66 608.
——, and E. G. Tabak, 2004: A Hamiltonian formulation for long
internal waves. Physica D, 195, 106–122.
2623
——, and N. Yokoyama, 2009: Nonlinear wave-wave interactions
in stratified flows: Direct numerical simulations. Physica D,
238, 803–815.
——, K. L. Polzin, and E. G. Tabak, 2004: Energy spectra of the
ocean’s internal wave field: Theory and observations. Phys.
Rev. Lett., 92, 128501, doi:10.1103/PhysRevLett.92.128501.
McComas, C. H., and F. P. Bretherton, 1977: Resonant interaction
of oceanic internal waves. J. Geophys. Res., 82, 1397–1412.
——, and P. Müller, 1981: Time scales of resonant interactions
among oceanic internal waves. J. Phys. Oceanogr., 11, 139–147.
Milder, D. M., 1990: The effects of truncation on surface-wave
Hamiltonian. J. Fluid Mech., 216, 249–262.
Müller, P., and D. J. Olbers, 1975: On the dynamics of internal
waves in the deep ocean. J. Geophys. Res., 80, 3848–3860.
——, ——, and J. Willebrand, 1978: The IWEX spectrum. J. Geophys. Res., 83, 479–500.
——, G. Holloway, F. Henyey, and N. Pomphrey, 1986: Nonlinear
interactions among internal gravity waves. Rev. Geophys., 24,
493–536.
Olbers, D. J., 1974: On the energy balance of small scale internal
waves in the deep sea. Hamburger geophysikalische Einzelschriften 27, 102 pp.
——, 1976: Nonlinear energy transfer and the energy balance of the
internal wave field in the deep ocean. J. Fluid Mech., 74, 375–399.
Pelinovsky, E. N., and M. A. Raevsky, 1977: Weak turbulence of
the internal waves of the ocean. Izv. Acad. Sci. USSR Atmos.
Oceanic Phys., 13, 187–193.
Polzin, K. L., 2004: A heuristic description of internal wave dynamics. J. Phys. Oceanogr., 34, 214–230.
——, E. Kunze, J. M. Toole, and R. W. Schmitt, 2003: The partition
of finescale energy into internal waves and subinertial motions. J. Phys. Oceanogr., 33, 234–248.
Pomphrey, N., J. D. Meiss, and K. M. Watson, 1980: Description of
nonlinear internal wave interactions using Langevin methods.
J. Geophys. Res., 85, 1085–1094.
Schmitt, R. W., J. R. Ledwell, E. T. Montgomery, K. L. Polzin, and
J. M. Toole, 2005: Enhanced diapycnal mixing by salt fingers in
the thermocline of the tropical Atlantic. Science, 308, 685–688.
Sherman, J. T., and R. Pinkel, 1991: Estimates of the vertical
wavenumber–frequency spectra of vertical shear and strain.
J. Phys. Oceanogr., 21, 292–303.
Silverthorne, K. E., and J. M. Toole, 2009: Seasonal kinetic energy
variability of near-inertial motions. J. Phys. Oceanogr., 39,
1035–1049.
Voronovich, A. G., 1979: Hamiltonian formalism for internal waves
in the ocean. Izv. Acad. Sci. USSR Atmos. Oceanic Phys., 15,
52–57.
Weller, R. A., K. L. Polzin, D. L. Rundnick, C. C. Eriksen, and
N. S. Oakey, 1991: Forced ocean response during the Frontal
Air-Sea Interaction Experiment. J. Geophys. Res., 96, 8611–
8638.
——, P. W. Furey, M. A. Spall, and R. E. Davis, 2004: The largescale context for oceanic subduction in the northeast Atlantic.
Deep-Sea Res. I, 51, 665–699.
Zakharov, V. E., 1967: The instability of waves in nonlinear dispersive media. Sov. Phys. JETP, 24, 740–744.
——, 1968: Stability of periodic waves of finite amplitude on the
surface of a deep fluid. J. Appl. Mech. Tech. Phys., 2, 190–194.
——, V. S. L’vov, and G. Falkovich, 1992: Kolmogorov Spectra of
Turbulence I: Wave Turbulence. Springer-Verlag, 246 pp.
Copyright of Journal of Physical Oceanography is the property of American Meteorological Society and its
content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's
express written permission. However, users may print, download, or email articles for individual use.